Semigroup identities of tropical matrices through matrix ranks

Abstract

We prove the conjecture that, for any $n$, the monoid of all $n \times n$ tropical matrices satisfies nontrivial semigroup identities. To this end, we prove that the factor rank of a large enough power of a tropical matrix does not exceed the tropical rank of the original matrix.

title = "Semigroup identities of tropical matrices through matrix ranks",

abstract = "We prove the conjecture that, for any $n$, the monoid of all $n \times n$ tropical matrices satisfies nontrivial semigroup identities. To this end, we prove that the factor rank of a large enough power of a tropical matrix does not exceed the tropical rank of the original matrix.",

keywords = "math.RA, math.CO, 20M05, 20M07, 20M30, 47D03",

author = "Zur Izhakian and Glenn Merlet",

note = "13 pages",

year = "2018",

month = jun,

day = "28",

language = "English",

publisher = "ArXiv",

type = "WorkingPaper",

institution = "ArXiv",

}

TY - UNPB

T1 - Semigroup identities of tropical matrices through matrix ranks

AU - Izhakian, Zur

AU - Merlet, Glenn

N1 - 13 pages

PY - 2018/6/28

Y1 - 2018/6/28

N2 - We prove the conjecture that, for any $n$, the monoid of all $n \times n$ tropical matrices satisfies nontrivial semigroup identities. To this end, we prove that the factor rank of a large enough power of a tropical matrix does not exceed the tropical rank of the original matrix.

AB - We prove the conjecture that, for any $n$, the monoid of all $n \times n$ tropical matrices satisfies nontrivial semigroup identities. To this end, we prove that the factor rank of a large enough power of a tropical matrix does not exceed the tropical rank of the original matrix.